Spectral approximations of the Stokes problem by divergence-free functions

1987 ◽  
Vol 2 (3) ◽  
pp. 195-226 ◽  
Author(s):  
F. Pasquarelli ◽  
A. Quarteroni ◽  
G. Sacchi-Landriani
Keyword(s):  
Stokes Problem ◽  
Spectral Approximations ◽  
Divergence Free

DOMAIN DECOMPOSITION APPROXIMATION TO A GENERALIZED STOKES PROBLEM BY SPECTRAL METHODS

1991 ◽  
Vol 01 (04) ◽  
pp. 501-515 ◽  
Author(s):  
CLAUDIO CARLENZOLI ◽  
PAOLA ZANOLLI
Keyword(s):  
Spectral Methods ◽  
Compressible Flows ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Navier Stokes ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Generalized Stokes Problem ◽  
Spectral Approximations ◽  
Viscous Compressible Flows

We consider here the approximation of a generalized Stokes problem by spectral methods in the collocation form. This problem is of particular interest when Navier-Stokes equations for viscous compressible flows are investigated. We also analyze a coupling of a viscous model with an inviscid one; precisely, we split the computational domain in two parts and in one of them we eliminate the viscous coefficient from the Stokes equations. Such an approach can be worthwhile in the study of compressible fluids around rigid profiles with critical layers. Finally we consider some numerical results with the aim of showing the excellent accuracy of the spectral approximations, as well as the efficiency of an iterative algorithm that we propose in order to alternate viscous and inviscid numerical solvers.

scholarly journals Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

2005 ◽  
Vol 75 (254) ◽  
pp. 533-564 ◽  
Author(s):  
Jesús Carrero ◽  
Bernardo Cockburn ◽  
Dominik Schötzau
Keyword(s):  
Stokes Problem ◽  
Divergence Free

scholarly journals Divergence free virtual elements for the stokes problem on polygonal meshes

2017 ◽  
Vol 51 (2) ◽  
pp. 509-535 ◽  
Author(s):  
Lourenco Beirão da Veiga ◽  
Carlo Lovadina ◽  
Giuseppe Vacca
Keyword(s):  
Stokes Problem ◽  
Polygonal Meshes ◽  
Divergence Free

scholarly journals Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem

2020 ◽  
Author(s):  
Derk Frerichs ◽  
Christian Merdon
Keyword(s):  
Stokes Problem ◽  
Two Dimensions ◽  
Momentum Balance ◽  
Test Functions ◽  
Virtual Test ◽  
Right Hand ◽  
Virtual Element Methods ◽  
Divergence Free ◽  
Virtual Element ◽  
The Right

Abstract Nondivergence-free discretizations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterized by large discretizations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretized in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretization via an $L^2$-best approximation does not preserve the divergence, and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness, another divergence-preserving reconstruction is suggested based on Raviart–Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

scholarly journals Guaranteed Upper Bounds For The Velocity Error Of Pressure-Robust Stokes Discretisations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
P.L. Lederer ◽  
C. Merdon
Keyword(s):  
Mixed Methods ◽  
Error Control ◽  
Stokes Problem ◽  
Upper Bounds ◽  
Type Result ◽  
Numerical Examples ◽  
Velocity Error ◽  
Divergence Free ◽  
Pressure Independent ◽  
Primal And Dual

Abstract This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one.

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